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Properties

Label 2-13-13.12-c13-0-3
Degree $2$
Conductor $13$
Sign $0.698 + 0.715i$
Analytic cond. $13.9400$
Root an. cond. $3.73363$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 131. i·2-s − 1.77e3·3-s − 9.07e3·4-s + 2.77e4i·5-s + 2.33e5i·6-s + 2.61e3i·7-s + 1.16e5i·8-s + 1.55e6·9-s + 3.64e6·10-s + 4.76e6i·11-s + 1.61e7·12-s + (−1.21e7 − 1.24e7i)13-s + 3.43e5·14-s − 4.92e7i·15-s − 5.90e7·16-s + 9.78e7·17-s + ⋯
L(s)  = 1  − 1.45i·2-s − 1.40·3-s − 1.10·4-s + 0.794i·5-s + 2.04i·6-s + 0.00839i·7-s + 0.157i·8-s + 0.974·9-s + 1.15·10-s + 0.810i·11-s + 1.55·12-s + (−0.698 − 0.715i)13-s + 0.0121·14-s − 1.11i·15-s − 0.880·16-s + 0.982·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.698 + 0.715i$
Analytic conductor: \(13.9400\)
Root analytic conductor: \(3.73363\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :13/2),\ 0.698 + 0.715i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.833261 - 0.351200i\)
\(L(\frac12)\) \(\approx\) \(0.833261 - 0.351200i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (1.21e7 + 1.24e7i)T \)
good2 \( 1 + 131. iT - 8.19e3T^{2} \)
3 \( 1 + 1.77e3T + 1.59e6T^{2} \)
5 \( 1 - 2.77e4iT - 1.22e9T^{2} \)
7 \( 1 - 2.61e3iT - 9.68e10T^{2} \)
11 \( 1 - 4.76e6iT - 3.45e13T^{2} \)
17 \( 1 - 9.78e7T + 9.90e15T^{2} \)
19 \( 1 + 1.43e6iT - 4.20e16T^{2} \)
23 \( 1 - 1.30e9T + 5.04e17T^{2} \)
29 \( 1 + 6.55e8T + 1.02e19T^{2} \)
31 \( 1 - 6.23e8iT - 2.44e19T^{2} \)
37 \( 1 - 2.16e10iT - 2.43e20T^{2} \)
41 \( 1 - 4.45e10iT - 9.25e20T^{2} \)
43 \( 1 - 1.29e10T + 1.71e21T^{2} \)
47 \( 1 + 8.80e10iT - 5.46e21T^{2} \)
53 \( 1 + 1.70e11T + 2.60e22T^{2} \)
59 \( 1 - 1.83e11iT - 1.04e23T^{2} \)
61 \( 1 - 4.23e11T + 1.61e23T^{2} \)
67 \( 1 - 5.57e11iT - 5.48e23T^{2} \)
71 \( 1 + 2.04e12iT - 1.16e24T^{2} \)
73 \( 1 - 2.15e12iT - 1.67e24T^{2} \)
79 \( 1 - 3.60e11T + 4.66e24T^{2} \)
83 \( 1 + 6.69e11iT - 8.87e24T^{2} \)
89 \( 1 - 6.83e12iT - 2.19e25T^{2} \)
97 \( 1 - 8.53e12iT - 6.73e25T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.90247812738523202056522800469, −14.99142316602998870046704037756, −12.85935906586316002441353531997, −11.86927267995549463616426661202, −10.81064061379175678528717656451, −9.932754828928255582851815311566, −6.91847493716484478048106229155, −4.98636468645185363232632128872, −2.93227198447657266137674717804, −0.974959406605782532091826313096, 0.61664747459213649909888340868, 4.88319494268489296138412955741, 5.78990925550426030945393677158, 7.19902653640115849049931263402, 8.970651552099695015454189983210, 11.14479812401733119166595902714, 12.57563623262739380100433341308, 14.32803116977430005735304946296, 15.96982973842176312595875118825, 16.79536012841395226985006745323

Graph of the $Z$-function along the critical line